[5e] Vengeance Paladin/Celestial Warlock advice

This is a character who returned home to find his family dead at the hands of undead, and swore vengeance. He sought guidance, and his prayers were answered by a Celestial who used to be a follower of Tyr but became disillusioned and singleminded after Tyr's death and now acts independently to purge the world of evil. He made a pact with the celestial, and sought out paladin training to be able to truly fight. He now considers it his mission to deliver the searing light that drives out darkness, in a rather literal sense. Mechanically he's a paladin who's good at melee combat and being a 'face', and also capable of effective magic ranged attacks, backup healing, and general utility with spells. His general fighting style will be heavy armor, quarterstaff (with shillelagh), and shield, backed by eldritch blast when at range.

What I'm looking for advice on: Anything I might tweak in the early progression to be more effective, anything I might have missed. Ideas on what to do from level 13 on - I had originally considered just sticking to paladin, but celestial warlock also looks attractive to me now. I'm going to be playing him in AL so am limited to PHB and Xanathar's guide for options.

Things I've considered and rejected: I could do just one level of hexblade to get the CHA to attacks and focus more on paladin, but that's not the concept I'm going for. Standard wisdom is to go to level 6 Paladin before multiclassing to get the multiattack and the aura, but I don't want to wait that long to get my concept going. Hold Person, Misty Step, and lesser restoration are warlock spells that I'd find useful, but I will get two of them as oath spells and the other as a paladin spell at 8th level. Conceivably I could skip Hex as a warlock spell and just use hunter's mark to get the same damage, but I really like 'disadvantage on one ability' feature of hex, especially as a melee character. Odd stats are deliberate, STR is 15 for heavy armor, CON is 15 expecting to get a +1 from reslience down the road.

A few issues you might be overlooking
#1 You are wanting to use multiple bonus action abilities. You have at least 4 I can see coming up. Hex, Polearm Mastery bonus action attack, and the oath of vengeance bonus action advantage buff, shillelagh. You are going to be stepping over yourself with bonus actions and rarely actually see their potential.

#2 Putting extra attack 3 levels back is a big deal. Polearm Mastery itself could have made up for that but your focus on other bonus action abilities is a bit of a problem.

My recommendations.

Level 2 Paladin Level 3 Warlock is a fine start given you are taking Polearm Mastery.
1. Don't worry about hex early. Instead just divine smite.
2. Only use shillelagh when you can prebuff without attacking. The bonus difference isn't enough to worry about (at least till +2 or +3 difference or higher)
3. Use Oath of Vengance as desired

I agree with that advice. I've been playing a warlock and have found that hex is more situational than it's usually treated in white room analysis, so I don't plan to use it all the time in the first place. On my warlock, it's not uncommon that I'll want to use concentration for things like Hold Person or Hunger of Hadar in a big fight, and for smaller fights I often don't want to blow a spell slot for a small boost on damage when the enemies won't last long anyway. While I sometimes use it as a damage boost, a lot of times the big reason I'm hexing is to give disadvantage on strength to help with grappling. It's also harder to keep up between fights than it is in the 'white room,' especially if (like this character) I'll be doing a lot of 'what does this do? Let's use a ritual spell to find out' casting. For this character, like you said most of the time spending a slot to get +2d8 or more on an attack is going to do more damage than +1d6 on multiple attacks.

I don't think Shillelagh will be that hard to use, because this character doesn't want to be the one to initiate melee combat. My typical opening move will be to stand in front of ranged types and use Eldritch Blast, hoping that the enemy will come to me so that I get the free reaction attack from Polearm mastery. That gives me a round or more where I'll have an empty bonus action that I can use for shillelagh (or VOE or hex). From levels 5-7 I actually do more damage at range than in melee unless I'm smiting so a lot of times I'll be happy to stand and deliver, and once I get the attack it's still not a weak combo.

If we drop hex for most the game and shileleah becomes a situational buff then all warlock levels really adds to you is some rituals and short rest spell slots and a very good ranged attack option.

Have you considered doing paladin 2 warlock 2 as your start? Ignore the damage bonus to eldritch blast for now. Having a ranged option is good but it’s not really going to compare to your melee options even if you take agonizing blast. Besides it’s still an option even without agonizing blast. Instead take 2 of the warlock at will invocations for added utility, silent image and detect magic at will are both fun and useful.. Then take paladin up to extra attack.

If the goal is to stand back and EB then what is polearm mastery really adding? A chance at a reaction attack and similar damage to EB If and obly IF something approaches you. IMO Booming blade makes a better backup to EB as it gets you close enough to Damage To EB but doesn’t require a feat or a bonus action. It also makes you a little stickier to help keep whatever approaches off the squishier back line pcs.

The goal for this character is not to stand back and EB all the time, that's just what happens to be most effective for sustained damage for levels 5-7 so I'll take advantage of it then. Booming Blade is not a legal option; since this is an AL character I only get PHB plus one other book. Booming Blade comes from SCAG while Celestial Warlock comes from Xanathar's, and while you can learn spells from other books by buying scrolls, you can't learn cantrips that way.

I don't see the benefit of taking two levels of warlock instead of three. I would get the straight paladin stuff one level earlier, but lose 2nd level short rest spell slots (better smite, at 8th level can fuel misty step or hold person), 2nd level warlock spells that cover things I'm not good at, a familiar, a stack of utility rituals plus guidance, thorn whip to force move enemies, shillelagh to let me use CHA instead of strength for attacks and up my weapon die to d8. Also I'll pretty much need to spend treasure points on a magic weapon, and I become MAD for ASIs, since I no longer get to use CHA to boost ranged attacks, melee attacks, and all saving throws. That's a lot to give up, and makes the character much weaker overall, since the basic ability score, feat, and weapon choices become non-optimal.

The goal for this character is not to stand back and EB all the time, that's just what happens to be most effective for sustained damage for levels 5-7 so I'll take advantage of it then. Booming Blade is not a legal option; since this is an AL character I only get PHB plus one other book. Booming Blade comes from SCAG while Celestial Warlock comes from Xanathar's, and while you can learn spells from other books by buying scrolls, you can't learn cantrips that way.

I don't see the benefit of taking two levels of warlock instead of three. I would get the straight paladin stuff one level earlier, but lose 2nd level short rest spell slots (better smite, at 8th level can fuel misty step or hold person), 2nd level warlock spells that cover things I'm not good at, a familiar, a stack of utility rituals plus guidance, thorn whip to force move enemies, shillelagh to let me use CHA instead of strength for attacks and up my weapon die to d8. Also I'll pretty much need to spend treasure points on a magic weapon, and I become MAD for ASIs, since I no longer get to use CHA to boost ranged attacks, melee attacks, and all saving throws. That's a lot to give up, and makes the character much weaker overall, since the basic ability score, feat, and weapon choices become non-optimal.

The only big thing I agree with the others on, here, is to not bother with Hex unless you're in a fight where you'll be ranged primarily. I'd go so far as to consider not bothering to learn it until later levels when it lasts 8 hours.

Polearm is the only Bonus Action you've got here that is going to be used most rounds. Having one round per combat, at most, where you can't use it because you used Shillelagh just isn't going to matter outside the proverbial "white room". Also, depending on how the game is run, it's likely that you will often be able to cast it before a fight starts. If nothing else, just use EB in the first round of combat, so you don't lose any damage output for that round.

There will be a few levels where you aren't as effective in melee unless you Smite. Then you'll get past those levels, get level 5 Paladin, and be golden.

All that said, I really wish they'd reprinted the SCAG cantrips in Xanathar's.

Polearm is the only Bonus Action you've got here that is going to be used most rounds. Having one round per combat, at most, where you can't use it because you used Shillelagh just isn't going to matter outside the proverbial "white room". Also, depending on how the game is run, it's likely that you will often be able to cast it before a fight starts. If nothing else, just use EB in the first round of combat, so you don't lose any damage output for that round.

Yes, this doesn't worry me. In my experience, the first round of combat often doesn't let you get in melee attacks anyway; enemies are far off, party members are in your way, you're waiting for something to happen, and so on. And it often takes a round to see if you want to do your maximum damage, it's generally not obvious on round one that this is a good time to burn out every spell slot maxing spike damage, so not smiting in round 1 really isn't a problem. And 'blast and buff' works nicely to set up to use the reaction from polearm mastery.

All that said, I really wish they'd reprinted the SCAG cantrips in Xanathar's.